《感性工程》课程教学课件（Kansei Engineering）02Day 2 Kansei Modeling

感性工程Kansei EngineeringDay 2: Kansei ModelingNovember21,2017YoshiteruNAKAMORl,ProfessorEmeritusJapanAdvanced Institute of ScienceandTechnology

Kansei Engineering Day 2: Kansei Modeling November 21, 2017 Yoshiteru NAKAMORI, Professor Emeritus Japan Advanced Institute of Science and Technology 感性工程

Today's Contents1.FuzzySetTheory模糊集理论2.Exercise2:FuzzyReasoning模糊推理3.PossibilityModels可能性模型4.KanseiEvaluationExperiment感性评价实验:Report:Personal ModelsCheerful快乐愁绪Melancholy1234567Possibilitymodels可能性模型Rating scale

1. Fuzzy Set Theory 2. Exercise 2: Fuzzy Reasoning 3. Possibility Models 4. Kansei Evaluation Experiment 感性评价实验 • Report: Personal Models Today’s Contents Melancholy Cheerful 1 2 3 4 5 6 7 Rating scale 愁绪 快乐 2 Possibility models 可能性模型 模糊集理论 可能性模型 模糊推理

暖味模糊1. Fuzzy Set Theory·Afuzzyset=Asetwiththeindefiniteboundary·Amembershipfunction=Afunction indicatingthedegreeofmembershiptoa setThepossibilitythatanelementbelongstoafuzzyset.Thepossibilitythat6belongsto(Example)Afuzzyset"About5"1"About5"is0.781Membershipfunction5Y29123456897Thisseemstobesimilartotheprobabilitydensityfunction,butthisis regardedas apossibilitydistributionfunctionAbout5isalsocalledafuzzynumber3

• A fuzzy set = A set with the indefinite boundary • A membership function = A function indicating the degree of membership to a set 1. Fuzzy Set Theory 曖昧 模糊 3 The possibility that an element belongs to a fuzzy set. (Example) A fuzzy set “About 5” 5 1 2 3 4 6 7 8 9 1 2 3 4 5 6 7 8 9 The possibility that 6 belongs to 1 “About 5” is 0.7 This seems to be similar to the probability density function, but this is regarded as a possibility distribution function. Membership function About 5 is also called a fuzzy number

Sets (in traditional mathematics)X:The setofpositive integersX: The entire set[1,2,3,4,5,6, ..?(theframeworkofconsideration)A:ThesetofpositiveevennumbersYAcXA: A subset of X2,4,6,...7 C 1.2,3,4,5,6, ...aA2EAaisamemberofA2 is a member of A3± AaisnotamemberofAα±A3 is not a member of AThe boundary is crisp.The characteristic function (binary)Thecharacteristicfunction(binary)if aeA1f.(a)f,(3)= 0f,(2)=1;if aA0

X : The entire set (the framework of consideration) A : A subset of X a is a member of A a is not a member of A The characteristic function (binary) Sets (in traditional mathematics) 4 A ⊂ X a∈ A a∉ A    ∉ ∈ = a A a A f A a 0, if 1, if ( ) X : The set of positive integers 1, 2, 3, 4, 5, 6, ⋯ A : The set of positive even numbers 2, 4, 6, ⋯ ⊂ 1, 2, 3, 4, 5, 6, ⋯ 2 is a member of A 3 is not a member of A The characteristic function (binary) 2∈ A 3∉ A f A (2) =1; f A (3) = 0 The boundary is crisp. A X a

A set whose boundaryis not clearAsetinmathematicshasacrispboundarythatis,anymemberiseitherinoroutofthe set.Therefore,the characteristicfunction takes the valueof o or 1.However,inthesociety,therearemanysetswhereboundariesarenot clear.A = Furniture set = (chair, table, closet, piano, refrigerator, bathtub)Introduceamembershipfunctionbyexpandingthecharacteristicfunction:μ,(x):ThedegreethatxbelongstoA.μA(table )=μA(chair) =μA (closet) =μ(piano) =从 (refrigerator) =μ(bathtub)

• A set in mathematics has a crisp boundary, that is, any member is either in or out of the set. Therefore, the characteristic function takes the value of 0 or 1. • However, in the society, there are many sets where boundaries are not clear. A set whose boundary is not clear 5 A = Furniture set = {chair, table, closet, piano, refrigerator, bathtub} Introduce a membership function by expanding the characteristic function: (x) µ A : The degree that x belongs to A. µ A ( chair ) = µ A ( table ) = µ A (closet ) = µ A ( piano) = µ A (refrigerator ) = µ A ( bathtub) = 0.8 0.7 1.0 0.3 0.2 0.1

The membership valuedepends on thecontextHD.E.Rumelhart:IntroductiontoHumanInformationProcessingPossibilityisNOTprobabilityJohnWiley&Sons,1977Thesumisnotnecessarily1THE CHTContext 1Context2Possibilitythatμ(H)=0.3μA(H)= 0.8this symbol is H.Possibilitythatthis symbol is A.μ(H)= 0.9μ(H)= 0.1

The membership value depends on the context 6 D.E.Rumelhart: Introduction to Human Information Processing. John Wiley & Sons, 1977. µ A ( ) = 0.3 µ H ( ) = 0.9 µ A ( ) = 0.8 µ H ( ) = 0.1 Context 1 Context 2 Possibility that this symbol is A. Possibility is NOT probability. The sum is not necessarily 1. Possibility that this symbol is H

Fuzzy sets (Fuzzy numbers).Traditional sets in mathematics are crisp sets, which are treated by binary logic(i.e.,Oor1).Seethefigureontheleftbelow.·However,forexample,a setoftemperatures of"Hot"cannotbehandled bybinarylogic.(Thereareindividualdifferencesorregionaldifferences.)Quantifysubjectivitybymembershipfunctions.11Aor(x)fuo (at)HotHot℃℃00262830262830Characteristicfunction (binary logic)Membershipfunction(multivaluedlogic)

• Traditional sets in mathematics are crisp sets, which are treated by binary logic (i.e., 0 or 1). See the figure on the left below. • However, for example, a set of temperatures of “Hot" cannot be handled by binary logic. (There are individual differences or regional differences.)  Quantify subjectivity by membership functions. Fuzzy sets (Fuzzy numbers) 7 0 1 28 ℃ 26 30 Hot 0 1 28 ℃ 26 30 Hot Characteristic function (binary logic) Membership function (multivalued logic) (x) (x) µ Hot fHot

Union and common set (in the case of crisp setsfs(x)X : The whole setf.(x)A, B : Two crisp subsets of XAUB : Union0xfaus(x)=max(f (x), fr(x))Thelargervalue of two functions at each point x.fAUB(x)AB : Intersection (Common set)fan(x)fans(x)= min(f (x), fe(x)0xThe smaller value oftwo functions at each pointx8

8 0 1 f (x) AB f (x) AB 0 1 f (x) A f (x) B A, B： Two crisp subsets of X X ： The whole set A B ： Union A B ： Intersection (Common set) f AB (x) = max{f A (x), fB (x)} f AB (x) = min{f A (x), fB (x)} Union and common set (in the case of crisp sets) x x The larger value of two functions at each point x. The smaller value of two functions at each point x

Union and intersection (in the case of fuzzy setsX : The whole setμg(x)μA(x)1A, B : Twofuzzy subsets of XAUB : Union0xμAuB(x) =max(u (x), μg(x)Thelargervalueof two functions at each point x.HAUB(x)AnB :IntersectionμAnB(x)=min(u (x), μg(x))μAn(x)x0Thesmallervalue oftwofunctions ateachpointx.9

9 0 1 (x) µ A (x) µ B 0 1 (x) µ AB (x) µ AB A, B : Two fuzzy subsets ofX X ： The whole set A B ： Union A B ： Intersection µ AB (x) = max{µ A (x),µ B (x)} µ AB (x) = min{µ A (x),µ B (x)} Union and intersection (in the case of fuzzy sets) x x The larger value of two functions at each point x. The smaller value of two functions at each point x

Complementary set (in the case of a crisp set)X : The whole setLawofLaw ofexcluded middlecontradictionACXA : Asubset of XAUA=X, ANA=ΦA : The complementary set of AAcXNegationofAfa(x)fa(x)1The characteristicfunctionofAHotHotfa(x)=1- f(x)℃0if xeA1.262830lo, if x@ACharacteristicfunction (binarylogic)10

Complementary set (in the case of a crisp set) 10 A ⊂ X A ⊂ X    ∉ ∈ = = − x A x A f x f x A A 0, if 1, if ( ) 1 ( ) A A = X , A A = φ 0 1 28 ℃ 26 30 Hot f (x) A f (x) A Hot X ： The whole set A ： A subset of X A ：The complementary set of A The characteristic function of A Law of excluded middle Negation of A Characteristic function (binary logic) Law of contradiction